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2 FORMAL RULES OF DERIVATION 1S1
(by (5.7.1) and (2.3.7)) that ||u|| = supdlaj,..., ||_MJ), which allows the

identification of $e(E; F) with the product H «£?(£; F_f), From the defini-

i=l

tion, it follows at once that u is the derivative of/at X_Q if and only if w_fis the derivative of / at X_Q for 1 < / ^ m.

Remark. Let E, F be complex Banach spaces, and E₀, F₀ the underlying
real Banach spaces. Then if a mapping / of an open subset A of E into F
is differentiable at a point x₀, it is also differentiable with the same deriv-
ative, when considered as a mapping of A into F₀ (a linear mapping of E
into F being also linear as a mapping of E₀ into F₀). But the converse is
not true, as the example of the mapping z -» z (complex conjugate) of C
into itself shows at once; as a mapping of R² into itself, u: z -» z (which can
be written (x, y) -> (x, —y)) is differentiable and has at each point a deriv-
ative equal to u, by (8.1.3); but u is not a complex linear mapping, hence
the result. We return to that question in Chapter IX (9.10.1).

When the mapping / of A into F is differentiable at every point of A,
we say that/is differentiable in A; the mapping x -*f'(x) = D/C*0 °f A
into «S?(E; F) will be written/' or D/and called the derivative off in A.

2. FORMAL RULES OF DERIVATION

(8.2.1) Let E, F, G be three Banach spaces, A an open neighborhood of X_Q e E,
fa continuous mapping of A into F, y_Q =/(x₀), B an open neighborhood of
j₀ in F, g a continuous mapping of B into G. Then iff is differentiable at
x₀ and g differentiable at j>₀, the mapping /z = #₀/ (which is defined and
continuous in a neighborhood of jc₀) is differentiable at x₀ , and we have

By assumption, given e such that 0<e< 1, there is an r > 0 such that,
for ||,s || < r and ||f || < r, we can write

sty* + 0 = #0

with Hoj.0)!! < eNI and ||o₂(OII <e||*||. On the other hand, by (8.1.1) and
(5.5.1), there are constants a, b such that, for any s and t,

" ^s ^and



	2 FORMAL RULES OF DERIVATION 1S1 (by (5.7.1) and (2.3.7)) that \|\|u\|\| = supdlaj,..., \|\|_MJ), which allows the m identification of $e(E; F) with the product H «£?(£; F_f), From the defini- i=l tion, it follows at once that u is the derivative of/at X_Q if and only if w_fis the derivative of / at X_Q for 1 < / ^ m.

	Remark. Let E, F be complex Banach spaces, and E₀, F₀ the underlying real Banach spaces. Then if a mapping / of an open subset A of E into F is differentiable at a point x₀, it is also differentiable with the same deriv- ative, when considered as a mapping of A into F₀ (a linear mapping of E into F being also linear as a mapping of E₀ into F₀). But the converse is not true, as the example of the mapping z -» z (complex conjugate) of C into itself shows at once; as a mapping of R² into itself, u: z -» z (which can be written (x, y) -> (x, —y)) is differentiable and has at each point a deriv- ative equal to u, by (8.1.3); but u is not a complex linear mapping, hence the result. We return to that question in Chapter IX (9.10.1). When the mapping / of A into F is differentiable at every point of A, we say that/is differentiable in A; the mapping x -f'(x)* = D/C0 °f A into* «S?(E; F) will be written/' or D/and called the derivative off in A.

	2. FORMAL RULES OF DERIVATION (8.2.1) Let E, F, G be three Banach spaces, A an open neighborhood of X_Q e E, fa continuous mapping of A into F, y_Q =/(x₀), B an open neighborhood of j₀ in F, g a continuous mapping of B into G. Then iff is differentiable at x₀ and g differentiable at j>₀, the mapping /z = #₀/ (which is defined and continuous in a neighborhood of jc₀) is differentiable at x₀ , and we have

	By assumption, given e such that 0<e< 1, there is an r > 0 such that, for \|\|,s \|\| < r and \|\|f \|\| < r, we can write

	sty* + 0 = #0 with Hoj.0)!! < eNI and \|\|o₂(OII <e\|\|\|\|. On the other hand, by (8.1.1) and (5.5.1), there are constants a, b* such that, for any s and t, " ^s ^and